Gödel's Proof

A very nice introduction and summary to Gödel's famous Incompleteness Theorem in which he proved any sufficiently complex formal system that can be represented within an arithmetic system has to be in some way incomplete. In short, he demonstrated that within a formal system there are statements that it will be capable of generating that must remain undecidable, as well as the existence of theorems which are true but which cannot be derived from the axiomatic statements of the formal system.

I highly recommend this if you are curious about Gödel's Theorem and have a basic understanding of set theory and formal logic. I think it does a great job taking you through the steps of the theorem and explaining things along the way, as well as explaining why it is important, as it points to the limits of formal systems and to the nature of their meta-statements, which cuts to the quick of mathematics; limiting certain notions about provability in maths all the way down to basic number theory and arithmetic.